Structured, compactly supported Banach frame decompositions of decomposition spaces

Abstract

$\newcommand{mc}[1]{\mathcal{#1}}$ $\newcommand{D}{\mc{D}(\mc{Q},L^p,\ell_w^q)}$ We present a framework for the construction of structured, possibly compactly supported Banach frames and atomic decompositions for decomposition spaces. Such a space $\D$ is defined using a frequency covering $\mc{Q}=(Q_i)_{i\in I}$: If $({\phi }_i{)}_i$ is a suitable partition of unity subordinate to $\mc{Q}$, then $\Vert g\Vert_{\D}:=\left\Vert\left(\Vert\mc{F}^{-1}(φ_i\hat{g})\Vert_{L^p}\right)_{i}\right\Vert_{\ell_w^q}$. We assume $\mc{Q}=(T_iQ+b_i)_{i}$, with $T_i\in \mathrm{GL}({\mathbb{R}}^d),b_i\in {\mathbb{R}}^d$. Given a prototype $\gamma$, we consider the system \[Ψ_{c}=(L_{c\cdot T_i^{-T}k}γ^{[i]})_{i\in I,k\in\Bbb{Z}^d}\text{ with }γ^{[i]}=|\det T_i|^{1/2}\, M_{b_i}(γ\circ T_i^T),\] with translation $L_x$ and modulation $M_{\xi }$. We provide verifiable conditions on $\gamma$ under which ${\Psi }_c$ forms a Banach frame or an atomic decomposition for $\D$, for small enough sampling density $c>0$. Our theory allows compactly supported prototypes and applies for arbitrary $p,q\in (0,\infty ]$. Often, ${\Psi }_c$ is both a Banach frame and an atomic decomposition, so that analysis sparsity is equivalent to synthesis sparsity, i.e. the analysis coefficients $(⟨f,L_{c\cdot T_i^{-T}k}{\gamma }^{[i]}⟩{)}_{i,k}$ lie in ${\ell }^p$ iff $f$ belongs to a certain decomposition space, iff $f={\sum }_{i,k}{c}_k^{(i)}\cdot L_{c\cdot T_i^{-T}k}{\gamma }^{[i]}$ with $(c_k^{(i)}{)}_{i,k}\in {\ell }^p$. This is convenient if only analysis sparsity is known to hold: Generally, this only yields synthesis sparsity w.r.t. the dual frame, about which often only little is known. But our theory yields synthesis sparsity w.r.t. the well-understood primal frame. In particular, our theory applies to $\alpha$-modulation spaces and inhom. Besov spaces. It also applies to shearlet frames, as we show in a companion paper.